MG205: Econometrics Theory and Applications
Topic 8: Exploiting Time Variation
London School of Economics and Political Science
February 9, 2026
Cross-Sectional Data Cannot Separate Heterogeneity from Treatment Effects
Panel Data Gives Us New Tools to Address Endogeneity
The problem
- Endogeneity: \(\text{Cov}(x_{it}, e_{it}) \neq 0\)
- Violation of Assumption 5 \(\Rightarrow\) no identification
- OLS gives biased estimates of the parameters of interest
- Cross-sectional data alone cannot fix this
Today
- Assume a particular error structure: \(e_{it} = \alpha_i + u_{it}\)
- With panel data, construct estimators invariant to \(\alpha_i\)
Following the same units over time enables new identification and estimation strategies.
Two Estimators Remove Unit-Level Unobserved Heterogeneity
First Differences and LSDV
First Differences (FD)
- Population model: \(y_{it} = \beta x_{it} + \alpha_i + u_{it}\)
- Subtract consecutive observations:
\[\Delta y_{it} = \beta \Delta x_{it} + \Delta u_{it}\]
- \(\alpha_i - \alpha_i = 0\): unobserved heterogeneity disappears
Least Squares Dummy Variables (LSDV)
- Include a dummy for each unit \(i\):
\[y_{it} = \beta x_{it} + \sum_{j=2}^{N} \gamma_j \mathbb{1}[i=j] + u_{it}\]
- The dummies absorb \(\alpha_i\)
- Equivalent to FD for \(T=2\) (we prove this later)
Exercise 1: Unobserved Heterogeneity Biases Cross-Sectional Estimates
Airline Fares Depend on Unobserved Route and Time Characteristics
Two Sources of Omitted Variable Bias
\[ \begin{aligned} \log(\text{fare})_{it} &= \beta_0 + \beta_1\log(\text{distance})_i + \beta_2\text{competition}_{it} + e_{it} \\ e_{it} &= \gamma_i + \delta_t + u_{it} \end{aligned} \]
- \(\gamma_i\): route-specific, time-invariant unobserved heterogeneity
- Business relationships
- Airport amenities
- \(\delta_t\): common time shocks
- Fuel prices
- Economic conditions
- \(u_{it}\) is idiosyncratic error
- We worry that \(\text{Cov}(\text{competition}_{it}, \gamma_i) \neq 0\)
- Model not identified
- \(\hat{\beta}\) might be biased
Controlling for Distance and Year Dummies Does Not Remove Route Heterogeneity
The Proposed Model Falls Short
Estimated model
\[\begin{align*} \widehat{\log(\text{fare})}_{it} &= \hat{\beta}_{0} + \hat{\beta}_{1}\log(\text{distance})_{i} \\ &+ \hat{\beta}_{2}\text{competition}_{it} \\ &+ \hat{\delta}_{1}\mathbb{1}[t=2007] \\ &+ \hat{\delta}_{2}\mathbb{1}[t=2012] \end{align*}\]
What remains in the error?
- Recall: \(e_{it} = \gamma_i + \delta_t + u_{it}\)
- The year dummies address common time trends (\(\delta_t\))
- \(\gamma_i\) remains in the error
Since \(\text{Cov}(\text{competition}_{it}, \gamma_i) \neq 0\), OLS is biased.
First-Differencing Eliminates Route-Level Unobserved Heterogeneity
The First-Difference Estimator
- First-difference operator: \(\Delta x_{it} = x_{it} - x_{it-1}\)
- Example: Take the USA–UK route. Subtract 2002 from 2007, and 2007 from 2012.
\[ \Delta\log(\text{fare})_{it} = \beta_2\Delta\text{competition}_{it} + \Delta\delta_t + \Delta u_{it} \]
\(\gamma_i - \gamma_i = 0\): time-invariant route characteristics disappear.
With year dummies for transition periods (2002–2007 base, 2007–2012):
\[ \widehat{\Delta\log(\text{fare})}_{it} = \hat\alpha + \hat\beta_2\Delta\text{competition}_{it} + \hat\delta\mathbb{1}[\text{transition } 2007-2012] \]
Combining FD with Year Dummies Addresses Both Sources
Two Strategies for Two-Way Heterogeneity
FD removes \(\gamma_i\) (unit FE)
- Subtract consecutive observations
- \(\gamma_i - \gamma_i = 0\)
- Time-invariant variables also drop out: \(\Delta\log(\text{distance})_i = 0\)
Year dummies absorb \(\delta_t\) (time FE)
- Include dummies for transition periods
- Common time shocks captured
- This is LSDV applied to time effects
(1) FD + year dummies, or (2) full LSDV with dummies for both units and time periods.
Rejecting the Null Does Not Validate the Model
The Trap
- With robust \(t\)-statistics, we reject \(H_{0}: \beta_{\text{competition}} = 0\)
- But the null assumes the model is correctly specified
- If OVB remains (route-level heterogeneity not addressed), \(\hat{\beta}\) is biased
- Statistical significance \(\neq\) valid causal interpretation
- The estimate is a linear projection
- FD reduces bias from time-invariant confounders but does not eliminate all sources
Exercise 2: Time Effects Capture Industry-Wide Patent Growth
Industry-Wide Patent Growth Requires Flexible Time Effects
Year Dummies Capture Any Form of Common Time Trends
Setting
- 37 pharmaceutical firms
- 2005-2007
- No OVB concerns
- Causal interpretation
- Patents growing industry-wide
- Regardless of individual firm R&D
Model
\[\begin{align*} \log(\text{patents})_{it} &= \beta_0 + \beta_1\log(\text{R\&D})_{it} \\ &+ \beta_2\mathbb{1}[t=2006] + \beta_3\mathbb{1}[t=2007] \\ &+ e_{it} \end{align*}\]
- Could model the trend linearly or quadratically
- Year dummies allow any form — nonparametric
- \(\beta_{1}\): elasticity of patents w.r.t. R&D (causal)
Year Dummy Coefficients Measure Growth Rates
Conditional Expectations Are The Tool to Interpret
\[\begin{align*} \mathbb{E}[\log(\text{patents})_{it} \mid t=2005] &= \beta_0 + \beta_1\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2006] &= (\beta_0 + \beta_2) + \beta_1\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2007] &= (\beta_0 + \beta_3) + \beta_1\log(\text{R\&D})_{it} \end{align*}\]
- Average log change in patents across all firms
- Geometric mean growth rate of patents in the industry, holding R&D constant
- \(\beta_2\): 2005 to 2006
- \(\beta_3\): 2005 to 2007
Year dummies measure growth rates between periods — not “the level in 2006 vs 2005.”
Interactions Allow the R&D Elasticity to Vary Over Time
Heterogeneous Elasticities
\[ \begin{align*} \log(\text{patents})_{it} &= \beta_0 + \beta_1\log(\text{R\&D})_{it} + \beta_2\mathbb{1}[t=2006] + \beta_3\mathbb{1}[t=2007] \\ &+ \beta_4(\log(\text{R\&D})_{it} \times \mathbb{1}[t=2006]) + \beta_5(\log(\text{R\&D})_{it} \times \mathbb{1}[t=2007]) \\ &+ e_{it} \end{align*} \]
Conditional means by year
\[\begin{align*} \mathbb{E}[\log(\text{patents})_{it} \mid t=2005] &= \beta_0 + \beta_1\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2006] &= (\beta_0 + \beta_2) + (\beta_1 + \beta_4)\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2007] &= (\beta_0 + \beta_3) + (\beta_1 + \beta_5)\log(\text{R\&D})_{it} \end{align*}\]
Level Shifts and Slope Shifts Are Separately Identified
Decomposing Differential Effects
| Year | Intercept | Elasticity |
|---|---|---|
| 2005 | \(\beta_0\) | \(\beta_1\) |
| 2006 | \(\beta_0 + \beta_2\) | \(\beta_1 + \beta_4\) |
| 2007 | \(\beta_0 + \beta_3\) | \(\beta_1 + \beta_5\) |
How the patents-R&D elasticity changes over time
- \(\beta_4\): elasticity change 2005 \(\to\) 2006
- \(\beta_5\): elasticity change 2005 \(\to\) 2007
No need to interpret \(\beta_4\) and \(\beta_5\) individually; the conditional means do the work.
Exercise 3: Empirical Models Interact All Relevant Variables
Gender Wage Gaps Changed After the Mining Boom
Three Patterns from the Data
- Gender: Men earn a constant wage premium over women
- Time trend: Wages grow over time for both groups
- Structural break (2005): After the mining company arrives, the male premium widens
The Empirical Model Interacts Gender, Time, and Post-2005
Eight Coefficients for Four Groups
\[ \begin{align*} \log(\text{wages})_{it} &= \beta_0 + \beta_1\mathbb{1}[i\text{ is male}] + \beta_2 t + \beta_3\mathbb{1}[t \geq 2005] \\ &+ \beta_4(\mathbb{1}[i\text{ is male}] \times t) + \beta_5(\mathbb{1}[i\text{ is male}] \times \mathbb{1}[t \geq 2005]) \\ &+ \beta_6(t \times \mathbb{1}[t \geq 2005]) + \beta_7(t \times \mathbb{1}[t \geq 2005] \times \mathbb{1}[i\text{ is male}]) \\ &+ e_{it} \end{align*} \]
This model captures level differences, trends, and how both changed after 2005, separately for men and women.
Conditional Means: Pre-2005
Women and Men Before the Structural Break
Women before 2005 (base category)
\[ \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=0,t<2005] = \beta_0 + \beta_2 t \]
Men before 2005
\[ \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=1,t<2005] = (\beta_0 + \beta_1) + (\beta_2 + \beta_4)t \]
\(\beta_1\) shifts the intercept; \(\beta_4\) shifts the slope.
Conditional Means: Post-2005
Women and Men After the Structural Break
Women after 2005
\[ \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=0,\; t \geq 2005] = (\beta_0 + \beta_3) + (\beta_2 + \beta_6)t\]
Men after 2005
\[\begin{align*} \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=1,\; t \geq 2005] &= (\beta_0 + \beta_1 + \beta_3 + \beta_5) \\ &\quad + (\beta_2 + \beta_4 + \beta_6 + \beta_7)t \end{align*}\]
Each coefficient modifies either the intercept or slope for a specific group-period combination.
Taking Differences Isolates Each Coefficient’s Role
Condition on Group, Then Difference
Coefficient Signs Follow Directly from the Figure
Economic Interpretation
Positive (\(> 0\))
- \(\beta_1\): male premium
- \(\beta_2\): wages grow over time
- \(\beta_7\): male wages grow faster post-2005
Zero (\(= 0\))
- \(\beta_3\): no level break for women at 2005
- \(\beta_4\): same pre-2005 growth rate
- \(\beta_6\): female growth unchanged post-2005
Negative (\(< 0\))
- \(\beta_5\): relative to the base group (women pre-2005), the intercept for men post-2005 is lower than what other coefficients predict
Exercise 4: Panel Data Enables Identification and Estimation
Panel Data Follows the Same Units Over Time
Definition and Structure
- \(y_{it}\), \(x_{it}\) for \(i = 1, \ldots, N\) and \(t = 1, \ldots, T\)
- Panel data: same units observed across multiple time periods
- Cross-section: one \(t\) only
- Repeated cross-section: same population, different individuals each period
Error decomposition
\[ e_{it} = \alpha_i + v_{it} \]
Panel Structure Enables Identification of Parameters
Identification vs Estimation
Identification
- Could we recover unique values for each parameter?
- Cross-section: \(\alpha_i\) in error, correlated with \(x_{it}\) \(\to\) cannot identify \(\beta\)
- Panel: difference out \(\alpha_i\)
- Requires: \(\mathbb{E}[v_{it} \mid x_{it}] = 0\)
Estimation
- Given identification, how do we compute \(\hat\beta\) from the data?
\[ \Delta y_i = \delta + \beta_1\Delta x_{i1} + \cdots + \beta_k\Delta x_{ik} + \Delta v_i \]
- Requires \(T \geq 2\) and within-unit variation
Example: cannot estimate returns to education via FD if education does not change over time.
Panel Data Reduces OVB but Cannot Eliminate All Sources of Bias
Solves
- Time-invariant OVB (\(\alpha_i\))
- Unit-invariant OVB (\(\lambda_t\))
Does not solve
- Time-varying confounders (\(u_{it}\))
- Measurement error
- Selection bias
Less scope for OVB, but not zero.
Exercise 5: First-Differencing Amplifies Measurement Error
Education Is Measured with Error in Both Periods
True vs Observed Variables
\[ \text{education}_{it} = \text{education}^{*}_{it} + e_{it} \]
Assumptions
- \(e_{it}\) uncorrelated with true education and other variables
- Education varies little over time for adults
Cross-Sectional Attenuation Bias Shrinks the Coefficient Towards Zero
The Baseline Problem
Population model
\[ \log(\text{wage})_i = \alpha + \beta\text{education}^{*}_i + \epsilon_i \]
Observed model substitutes \(\text{education}_i = \text{education}^{*}_i + e_i\)
Attenuation bias
\[ \text{plim}\;\hat\beta = \beta \cdot \frac{\text{Var}(\text{educ}^*)}{\text{Var}(\text{educ}^*) + \text{Var}(e)} \]
The ratio is less than 1, so the coefficient is biased towards zero (derived in Topic 6).
First-Differencing Increases the Measurement Error Variance
The Panel Data Paradox
FD of observed education
\[ \Delta\text{education}_i = \Delta\text{education}^{*}_{i} + (e_{i2} - e_{i1}) \]
If \(e_{i1}\) and \(e_{i2}\) are uncorrelated
\[ \text{Var}(e_{i2} - e_{i1}) = \text{Var}(e_{i1}) + \text{Var}(e_{i2}) \]
Panel Data Involves a Fundamental Bias Trade-off
FD Eliminates Fixed Confounders but Amplifies Measurement Error
\[ \text{plim}\;\hat\beta_{\text{FD}} = \beta \cdot \frac{\text{Var}(\Delta\text{educ}^*)}{\text{Var}(\Delta\text{educ}^*) + \text{Var}(e_{i1}) + \text{Var}(e_{i2})} \]
Benefits
- Eliminates \(\alpha_i\), reduces OVB from time-invariant confounders
- Enables causal identification under strict exogeneity
Costs
- Measurement error variance grows in denominator
- Numerator small if \(x\) changes little over time
- Attenuation ratio is smaller — more severe bias towards zero
Exercise 6: Panel Data Cannot Solve Selection Bias
Roommate Nationality May Affect Student Grades
Self-Selection into Rooms Creates Endogeneity
Let \(\text{same}_{it} = \mathbb{1}[i\text{ has same-nationality roommate in } t]\)
\[ \text{grades}_{it} = \alpha + \beta\;\text{same}_{it} + e_{it} \]
- Exogeneity holds under random assignment of roommates
- If students choose roommates: an omitted equation determines room selection
- More outgoing students may prefer different nationalities and perform differently academically
- \(\text{Cov}(\text{same}_{it}, e_{it}) \neq 0\) arises from selection, not unobserved heterogeneity
Two Years of Data Require Roommate Changes
Panel Structure and Exogenous Mobility
Year 1
\(\text{grades}_{i1} = \alpha + \beta\;\text{same}_{i1} + \alpha_i + u_{i1}\)
Year 2
\(\text{grades}_{i2} = \alpha + \beta\;\text{same}_{i2} + \alpha_i + u_{i2}\)
First-differencing
\(\Delta\text{grades}_i = \beta\;\Delta\text{same}_i + \Delta u_i\)
- Critical: students must change roommates (\(\Delta\text{same}_i \neq 0\) for some)
- Exogenous mobility design — plausible if the university reassigns rooms
The Problem Is Selection, Not Unobserved Heterogeneity
What Panel Data Cannot Fix
- FD removes \(\alpha_i\) (unobserved ability) — but the core problem is selection into rooms
- If reasons for changing roommates correlate with grade changes, FD does not help
- Panel data addresses unobserved heterogeneity
- It does not address selection bias
Random Assignment Plus Panel Data Strengthens Identification
You Get What You Pay For
- Random assignment of roommates:
- \(\beta\) is identified even in cross-section
- Panel adds precision: removes \(\alpha_i\) from the error \(\to\) smaller variance \(\to\) smaller standard errors
- Potential selection: panel data alone cannot solve the selection problem
Inference, Functional Forms, and Composition in Panel Data
We Run into an Old Friend!
Inference
- AS2 violated: same unit observed repeatedly
- AS7 may fail: heterogeneous units
- Affects SEs, not \(\hat{\beta}\)
Functional Forms
- Linear vs nonparametric time trends
- Treatment absorbed by time FE
- Unbalanced panels → composition effects
Exercise 7: Clustered Standard Errors Account for Within-Unit Correlation
Clustering Allows Arbitrary Within-Unit Correlation
A Conservative Fix for Panel Inference
What clustering does
- Allows arbitrary correlation within cluster
- Requires independence across clusters
- Does not change \(\hat{\beta}\) — only the standard errors
Two sources of correlation
- Within-unit persistence: worker earnings persist year to year
- Within-group spillovers: departmental training affects all workers
Serial Correlation Only Affects Inference
The Three Pillars: Identification, Estimation, Inference
- Identification (AS1, AS3-AS5): Can we recover \(\beta\)? Not affected
- Estimation (AS2): How do we compute \(\hat{\beta}\)? Not affected
- Inference (AS2, AS7): Are our SEs, p-values, CIs valid?
- Assumption 2 (i.i.d. sampling): same unit observed repeatedly → errors correlated within unit
- Assumption 7 (homoskedasticity): error variance may differ across units
- Both violations affect inference only — \(\hat{\beta}\) unchanged, SEs wrong
Exercise 8: Time Fixed Effects Can Be Collinear with Treatment
Measuring the Effect of Increased Force
All Municipalities Treated Simultaneously
Empirical setup
- Dependent variable: Drug usage at municipality \(i\) on day \(t\)
- Treatment: All municipalities increase police on the same date (vertical line in figure)
- Drug usage has a pre-existing upward trend
Day Fixed Effects Cannot Separate Treatment from Time
The Collinearity Trap
Ideal model: \(\text{drug usage}_{it} = \mu\;\text{post}_t + \theta_i + \rho_t + e_{it}\)
Conditional expectations — with \(T = 4\), treatment at \(t = 3\):
\[\begin{align*} \mathbb{E}[\text{drug usage}_{it} \mid t=1] &= \theta_i + \rho_1 \\ \mathbb{E}[\text{drug usage}_{it} \mid t=2] &= \theta_i + \rho_2 \\ \mathbb{E}[\text{drug usage}_{it} \mid t=3] &= \mu + \theta_i + \rho_3 \\ \mathbb{E}[\text{drug usage}_{it} \mid t=4] &= \mu + \theta_i + \rho_4 \end{align*}\]
- \((\mu + \rho_3)\) observed jointly, not separately \(\Rightarrow\) \(\mu\) not identified
- Reason: \(\text{post}_t\) is a linear combination of day dummies
A Linear Time Trend Restores Identification
Parametric but Estimable
\[\text{drug usage}_{it} = \mu\text{post}_t + \theta_i + \gamma t + e_{it}\]
Interpretation of \(\gamma\)
- Average change per time unit
- Imposes linearity — may miss curvature
- Intermediate: polynomial
The trade-off
| Parametric trend | Day FE | |
|---|---|---|
| Flexibility | Low (linear) | High (any shape) |
| Estimate \(\mu\)? | Yes | No (collinear) |
| Risk | Misspecified trend | No identification |
When treatment varies only at the time level, time FE absorb it completely.
Exercise 9: Age Dummies Provide Nonparametric Functional Forms
When We Do Not Know the Functional Form, Use Dummies
Nonparametric Estimation
\[ \log(\text{earnings})_{it} = \alpha_i + \theta_t + \sum_{j=17}^{85} \gamma_j\;\mathbb{1}[\text{age}_{it} = j] + e_{it} \]
- \(\alpha_i\): individual FE (absorbs ability, education, etc.)
- \(\theta_t\): time FE (absorbs aggregate macroeconomic trends)
- \(\gamma_j\): average difference in log-earnings between workers aged \(j\) and workers aged 16, holding constant \(\alpha_i\) and \(\theta_t\)
- Nonparametric: no assumption on the shape of the age-earnings relation
- Parametric: quadratic requires \(f(\text{age}) = \beta_1\;\text{age} + \beta_2\;\text{age}^2\)
Imprecision at Extremes Reflects Thin Data
The Variance Formula for Dummy Variables
Each age dummy \(d_j = \mathbb{1}[\text{age}_{it} = j]\) is a binary variable with proportion \(p_j = n_j/n\):
\[ \text{Var}(\hat{\gamma}_j) \propto \frac{\sigma^2}{n \cdot p_j(1 - p_j)} \]
- Most workers aged 25-64 \(\Rightarrow\) \(p_j\) near its maximum \(\Rightarrow\) \(\text{Var}(\hat{\gamma}_j)\) small
- Few workers at ages 16-24 and 65-85 \(\Rightarrow\) \(p_j \approx 0\) \(\Rightarrow\) \(\text{Var}(\hat{\gamma}_j)\) large
- \(\hat{\gamma}_j\) at extreme ages has wide confidence intervals
Nonparametric flexibility comes at the cost of imprecision where data is thin.
Exercise 10: Fixed Effects Decompose Treatment Into Incentive and Selection
Performance Pay Increases Firm-Level Productivity by 20%
But How Much Is Incentives vs Selection?
A firm introduces performance pay. The panel is unbalanced: some workers leave (exiters), some stay (stayers), some join (entrants).
\[ \log(\widehat{\text{productivity}})_{it} = \hat{\alpha}_i + \hat{\beta}\;\text{performance pay}_t \]
OLS (Pooled)
- \(\hat{\beta}_{\text{OLS}} = 0.20\) (SE \(= 0.03\))
- Captures total effect at firm level
- Uses all workers (stayers + exiters + entrants)
Fixed Effects
- \(\hat{\beta}_{\text{FE}} = 0.10\) (SE \(= 0.02\))
- Captures within-worker incentive effect only
- Only stayers contribute to identification
Half the Effect Is Incentives, Half Is Composition
The Decomposition
Incentive effect
- \(\hat{\beta}_{\text{FE}} = 0.10\)
- Same workers produce more under performance pay
- They work harder
Composition effect
- \(\hat{\beta}_{\text{OLS}} - \hat{\beta}_{\text{FE}} = 0.20 - 0.10 = 0.10\)
- Different workers join the firm under performance pay
- \(\mathbb{E}[\alpha_i \mid \text{entrant}] > \mathbb{E}[\alpha_i \mid \text{exiter}]\)
OLS captures total change; FE isolates the within-unit mechanism. The difference is the selection channel.
Exercise 11: Worker Composition Changes Require Fixed Effects
This Exercise Combines Three Previous Challenges
Collinearity (Q8) + Composition (Q10) + Time-Varying Controls (New)
\[ \text{productivity}_{it} = \beta_1\;\text{contingent}_t + \beta_2\;\text{weather}_t + \beta_3\;\text{width}_{it} + \beta_4\;\text{height}_{it} + e_{it} \]
Q8 callback: Collinearity with time FE
- Contingent pay switches on one date for everyone
- \(\text{contingent}_t\) collinear with day dummies
- Cannot include time FE → use weather as observable time-varying control
Q10 callback: Negative composition
- Best workers leave for blueberry harvest (neighbouring farms offer better alternatives)
- Remaining workers less productive: \[\mathbb{E}[\gamma_i \mid \text{second half}] < \mathbb{E}[\gamma_i \mid \text{first half}]\]
- Direction flips from Q10 (where better workers joined)
Width and Height Are Time-AND-Individual Varying
Controls vs Fixed Effects Address Different Problems
\[\begin{align*} \text{productivity}_{it} &= \beta_1\;\text{contingent}_t + \beta_2\;\text{weather}_t + \beta_3\;\text{width}_{it} + \beta_4\;\text{height}_{it} \\ &+ \gamma_i + e_{it} \end{align*}\]
- Width and height vary across workers AND days — not captured by worker FE (\(\gamma_i\) absorbs time-invariant traits only)
- Still susceptible to Assumption 5: if correlated with \(e_{it}\), OVB persists
- Only workers in both periods contribute — churn shrinks effective sample
Controls handle time-varying confounders; FE handles time-invariant heterogeneity.
Exercise 12: Multiple Fixed Effects Identify Leader Quality Through Rotation
Call Centre Productivity Varies Across Operators, Leaders, and Days
Three Sources of Variation
Setting
- Outcome: calls answered per hour
- Panel of operators \(\times\) days
- Random call assignment: no operator-call selection
- Operators and leaders rotate across teams
- Rotation provides identifying variation
Three sources of heterogeneity
- \(\lambda_i\): operator ability
- \(\mu_j\): leader quality (the parameter of interest)
- \(\theta_t\): day-level conditions (holidays, system outages)
Three-Way Fixed Effects Isolate Leader Quality
Rotation Is the Identification Condition
\[ \text{productivity}_{ijt} = \lambda_i + \mu_j + \theta_t + e_{ijt} \]
- \(\lambda_i\): operator FE — absorbs intrinsic worker ability
- \(\mu_j\): leader FE — the object of interest
- \(\theta_t\): day FE — absorbs common daily shocks
- Identification requires rotation: same operator observed under different leaders
- Without rotation, \(\lambda_i\) and \(\mu_j\) are not separately identified
The F-Test Provides Evidence for Leader Quality Differences
Joint Significance of Leader Fixed Effects
Restricted vs unrestricted
- Restricted (\(H_0\) true): \(\text{productivity}_{ijt} = \lambda_i + \mu + \theta_t + e_{ijt}\)
- Unrestricted (\(H_1\)): \(\text{productivity}_{ijt} = \lambda_i + \mu_j + \theta_t + e_{ijt}\)
Five-step framework
- Choose \(\alpha = 0.05\)
- \(H_0: \mu_2 = \mu_3 = \cdots = \mu_J = 0\)
- \(F = \frac{(\text{RSS}_R - \text{RSS}_U)/q}{\text{RSS}_U/(n - K)}\;,\quad q = J - 1\)
- Reject \(H_0\) if \(F > F_{q,\; n-K,\; \alpha}\)
- If reject: leaders differ in quality
Multiple fixed effects require sufficient rotation across dimensions for identification.
Panel Data Provides Identification Through Within-Unit Variation
Summary (I)
- FD and LSDV address different components of unobserved heterogeneity
- Time effects capture common growth; interactions capture heterogeneous effects
- Measurement error is amplified by first-differencing — a fundamental trade-off
- Panel data cannot solve selection bias
- Clustered standard errors correct for within-unit serial correlation
Panel Data Provides Identification Through Within-Unit Variation
Summary (II)
- Time fixed effects can be collinear with treatment
- Dummies provide nonparametric functional forms at the cost of imprecision at extremes
- Fixed effects decompose total effects into within-unit incentive and composition channels
- Multiple fixed effects require rotation for identification; F-tests assess joint significance
Next Week: Discrimination
Empirical Exercise 6
Bertrand & Mullainathan (2004)
- Are Emily and Greg more employable than Lakisha and Jamal?
- Resume audit study
- Randomly assigned racial signals
- Experimental identification
- Randomisation breaks OVB
- Asymmetric returns to resume quality by race
Appendix: Derivations and Extensions
First-Difference Derivation for General Panel Model
Algebra
Write the model for \(t = 1\) and \(t = 2\):
\[\begin{align*} y_{i1} &= \beta_0 + \beta_1 x_{i1,1} + \cdots + \beta_k x_{i1,k} + a_i + v_{i1} \\ y_{i2} &= (\beta_0 + \delta) + \beta_1 x_{i2,1} + \cdots + \beta_k x_{i2,k} + a_i + v_{i2} \end{align*}\]
Subtract: \(a_i - a_i = 0\).
\[ \Delta y_i = \delta + \beta_1\Delta x_{i1} + \beta_2\Delta x_{i2} + \cdots + \beta_k\Delta x_{ik} + \Delta v_i \]
First-Differencing Removes Time-Invariant Unobservables
- \(a_i\) eliminated by differencing: \(a_i - a_i = 0\)
- OLS on \(\Delta y_i\) is consistent under strict exogeneity: \(\mathbb{E}[\Delta v_i \mid \Delta x] = 0\)
- Time-invariant variables (e.g., distance, gender) also drop out
- Generalises to \(T > 2\): take consecutive differences for each pair \((t, t-1)\)
Attenuation Factor: Cross-Section vs First-Differences
Cross-section
\[ \text{plim}\hat\beta_{\text{CS}} = \beta \cdot \frac{\text{Var}(\text{educ}^*)}{\text{Var}(\text{educ}^*) + \text{Var}(e)} \]
First-differences (assuming uncorrelated measurement errors)
\[ \text{plim}\;\hat\beta_{\text{FD}} = \beta \cdot \frac{\text{Var}(\Delta\text{educ}^*)}{\text{Var}(\Delta\text{educ}^*) + \text{Var}(e_{i1}) + \text{Var}(e_{i2})} \]
Why FD amplifies the bias
- Numerator shrinks: education barely changes → \(\text{Var}(\Delta\text{educ}^*) \ll \text{Var}(\text{educ}^*)\)
- Denominator doubles: \(\text{Var}(e_{i1}) + \text{Var}(e_{i2}) = 2\text{Var}(e)\)
Why Log?
Three Reasons for Using Log-Earnings
- Skewness: raw earnings are right-skewed
- \(\log(\text{earnings})\) yields a more symmetric distribution, closer to normality (supporting AS7)
- Multiplicative relationships: if earnings = base \(\times\) skill premium \(\times\) experience premium, then
- \(\log(\text{earnings}) = \log(\text{base}) + \log(\text{skill premium}) + \log(\text{experience premium})\)
- Log linearises multiplicative structures into additive ones
- Growth rates: \(\Delta\log(\text{earnings}) \approx \%\Delta\text{earnings}\)
- Differences in logs approximate percentage changes
- The natural unit for comparing workers across different baseline earnings
Age-Period-Cohort Identification Problem
Why Age, Time, and Individual FE Cannot All Be Included
The collinearity
\[ \log(\text{earnings})_{it} = \alpha_i + \theta_t + \sum_{j=17}^{85} \gamma_j\;\mathbb{1}[\text{age}_{it} = j] + e_{it} \]
suffers from a fundamental identification problem:
\[ \text{age}_{it} = \text{year}_t - \text{birth year}_i \]
- \(\text{birth year}_i\) is a linear function of \(\alpha_i\) (time-invariant)
- \(\text{year}_t\) is captured by \(\theta_t\)
- Therefore age is perfectly collinear with \(\alpha_i\) and \(\theta_t\)
OLS vs FE Decomposition — Formal
Stayers (\(S\)), Exiters (\(X\)), Entrants (\(N\))
Algebra
\[\begin{align*} \hat{\beta}_{\text{OLS}} &= \bar{y}_{\text{post}} - \bar{y}_{\text{pre}} \\ &= \left[\frac{|S|}{|S|+|N|}\bar{y}^S_{\text{post}} + \frac{|N|}{|S|+|N|}\bar{y}^N_{\text{post}}\right] \\ &\quad - \left[\frac{|S|}{|S|+|X|}\bar{y}^S_{\text{pre}} + \frac{|X|}{|S|+|X|}\bar{y}^X_{\text{pre}}\right] \end{align*}\]
\[ \hat{\beta}_{\text{FE}} = \frac{1}{|S|}\sum_{i \in S}(y_{i,\text{post}} - y_{i,\text{pre}}) \]
Interpretation
\(|S|\), \(|X|\), \(|N|\) = number of stayers, exiters, entrants.
- \(\hat{\beta}_{\text{OLS}}\): all workers — total effect
- \(\hat{\beta}_{\text{FE}}\): stayers only — incentive effect
- Not bias — different quantities
AKM: Matched Employer-Employee Framework
Abowd, Kramarz, and Margolis (1999, Econometrica)
The call centre exercise (Q12) builds on Abowd et al. (1999) and Fenizia (2022).
\[ \log(\text{wages})_{it} = \alpha_i + \psi_{J(i,t)} + x'_{it}\beta + e_{it} \]
- \(\alpha_i\): worker FE (ability, human capital)
- \(\psi_{J(i,t)}\): firm FE for the firm \(J\) employing worker \(i\) at time \(t\)
- Identification requires worker mobility across firms (exogenous mobility)
- Same logic as Q12: operators rotate across teams
- Without mobility, \(\alpha_i\) and \(\psi_{J(i,t)}\) not separately identified
Why Day FE and Post Are Collinear
A Concrete Example with Four Days
Treatment starts on day 3:
| Day | \(d_2\) | \(d_3\) | \(d_4\) | \(\text{post}_t\) |
|---|---|---|---|---|
| 1 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 |
| 3 | 0 | 1 | 0 | 1 |
| 4 | 0 | 0 | 1 | 1 |
\(\text{post}_t = d_3 + d_4\) — an exact linear combination of the day dummies. Stata would drop one variable automatically. The treatment effect \(\mu\) cannot be separated from the day effects.
Replacing Day FE with a Parametric Trend Restores Identification
- Resolution: replace \(T-1\) day dummies with a single parametric trend (\(\gamma\;\text{time}_t\))
- Fewer parameters → no collinearity → \(\mu\) is identified
- Q8 application: all municipalities receive increased police force simultaneously — \(\text{post}_t\) is perfectly collinear with day FE
- Q11 application: contingent pay switches on a single date for all workers — same collinearity
- Trade-off: a linear trend is restrictive (assumes constant time effect) but estimable; day FE are flexible but break identification when treatment is universal